Calculus Course Overview

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Damon Cain
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1 Description: Walk in the footsteps of Newton and Leibnitz! An interactive text and graphing software combine with the exciting on-line course delivery to make Calculus an adventure. This course includes a study of limits, continuity, differentiation, and integration of algebraic, trigonometric and transcendental functions, and the applications of derivatives and integrals. Prerequisites: Algebra I, Geometry, Algebra II, Pre-Calculus or Trigonometry/Analytical Geometry. Estimated Completion Time: 2 Semesters Major and Concepts: Module 0: Preparation for Calculus Suggested Pace: 2 weeks * Understanding the properties of real numbers and the number line * Using the Cartesian coordinate system to graph functions * Comparing relative magnitudes of functions contrasting exponential, logarithmic and polynomial growth * Orientation to course * Real numbers and the real number line * Cartesian plane * Graphs and models * Linear models and rates of change * Functions and their graphs * Entry Quiz * Oral Review: Discussion about using Calculator zoom features to examine a graph in a good viewing window and calculator operations to find the zeros of a graph and the point of intersection of two graphs * Quiz Functions, Graphs, and Rates of Change Module 1: Limits and Continuity Suggested Pace: 2 weeks * Intuitive understanding of limit process * Calculating limits using algebraic methods * Estimating limits using tables of data * Estimating limits using graphs Page 1 of 7

2 * Understanding asymptotes graphically * Describing asymptotic behavior in terms of limits involving infinity * Intuitive understanding of continuity * Understanding continuity in terms of limits * Understanding graphs of continuous or non-continuous functions geometrically * Preview of calculus * Finding limits graphically and numerically * Evaluating limits analytically * Continuity and one-sided limits * Infinite limits * Problems sets * Quiz Calculating Limits * Oral Review: Discussion about using the Calculator to experiment and produce a table of values to examine a function and estimate a limit as x approaches a point and as x grows without bound. Discussion about the limitation of a graphing calculator to show discontinuities in functions and the value of using a calculator to support conclusions found analytically. * Elluminate Session: Discussion about conditions of continuity. * Test Limits and Continuity Module 2: Differentiation Suggested Pace: 5 weeks * Derivative defined as the limit of the difference quotient * Graphic, numeric and analytic interpretations of the derivative * Knowledge of derivatives of power and trigonometric functions * Basic rules for the derivatives of sums, products, and quotients of functions * Derivative interpreted as instantaneous rate of change * Continuity and differentiability * Slope of curve at a point * Tangent line to a curve at a point * Local linear approximation * Instantaneous rate of change as the limit of average rate of change * Approximate rate of change from graphs and tables of values * Chain rule and implicit differentiation * Equations involving derivatives and problems using their verbal descriptions * Modeling rates of change and solving related rates problems Page 2 of 7

3 * The derivative and the tangent line problem * Basic differentiation rules and rates of change * The product and quotient rules * The chain rule * Implicit differentiation * Related rates * Problem sets * Quiz Definition and computation of derivatives * Oral Review: Discussion about using a calculator to find the value of a derivative at a point, and how to graph the derived function using a calculator. Discussion about the limitations of the calculator to find the numerical derivative (for example, f (0) for f (x) = x ). * Test Differentiation Module 3: Applications of Differentiation Suggested Pace: 6 weeks * Corresponding characteristics of graphs of f and f * Relationship between the increasing and decreasing behavior of f and the sign of f * Corresponding characteristics of graphs of f, f, and f * Relationship between the concavity of f and the sign of f * Points of inflection as places where concavity changes * Mean Value Theorem and geometric consequences * Analysis of curves including monotonicity and concavity * Optimization absolute and relative extrema * Equations involving derivatives and problems using their verbal descriptions * Extrema on an interval * Rolle s Theorem and the Mean Value Theorem * Increasing and decreasing functions * Concavity and the second derivative test * Limits at infinity * Curve sketching * Optimization * Differentials Page 3 of 7

4 * Problem sets * Quiz Extrema and Concavity * Oral Review: Discussion about using the calculator to find the critical values of a function by examining the graph of the function and the graph of the function s derivative. * Test Applications of Derivatives * Semester Exam Module 4: Integration Suggested Pace: 4 weeks * Definite integral as a limit of Riemann sums * Definite integral of the rate of change of a quantity over an interval interpreted as the change of the quantity over the interval: * Basic properties of definite integrals * Use of the Fundamental Theorem of Calculus to evaluate definite integrals * Use of the Fundamental Theorem of Calculus to represent a particular antiderivative, and the analytical and graphical analysis of functions so defined * Find antiderivatives including the use of substitution * Finding specific antiderivatives using initial conditions, including applications to motion along a line * Use of Riemann sums and trapezoidal sums to approximate definite integrals of functions represented algebraically, graphically and by tables of values * Antiderivatives and Indefinite Integration * Area * Riemann sums and definite integrals * The Fundamental Theorem of Calculus * Integration by substitution * Numerical integration * Application of definite integrals including area, volume, position/velocity/acceleration and accumulation functions * The Integral as a function * Quiz Integration and Area * Quiz The Fundamental Theorem of Calculus * Oral Review: Discussion about using the calculator to estimate the value of a definite integral and to support solutions derived analytically. * Test Integration Page 4 of 7

5 Module 5: Transcendental Functions Suggested Pace: 3 weeks * Use of implicit differentiation in finding the derivative of the inverse of a function * Geometric interpretation of differential equations via slope fields * Relationship between slope fields and solution curves for differential equations * Knowledge of derivatives of exponential, logarithmic, and inverse trigonometric functions * Basic properties of definite integrals * Use of the Fundamental Theorem of Calculus to evaluate definite integrals * Find antiderivatives including the use of substitution * Application of integrals * The natural logarithmic function and differentiation * The natural logarithmic function and integration * Inverse functions including the relationship between the derivative of a function and its inverse at a point * Exponential functions * Bases other than e and applications * Inverse trigonometric functions and differentiation * Inverse trigonometric functions and integration * Quiz Natural Logarithmic Functions and Exponential Functions * Oral Review: Examine the limitations of the graphing calculator in graphing Natural Log functions. Students are required to verbally express the concepts related to the derivatives and integrals of exponential, logarithmic, and inverse trigonometric functions. * Test: Transcendental Functions Module 6: Differential Equations and Slope Fields: Suggested Pace: 3 weeks * Differential equations: growth and decay * Differential equations: separation of variables * Slope fields * Solving separable differential equations and using them in modeling * Geometric interpretation of differential equations via slope fields * Relationship between slope fields and solution curves for differential equations Page 5 of 7

6 * Oral Review: Examine the limitations of the graphing calculator in graphing Natural Log functions. Students are required to verbally express the concepts related to the derivatives and integrals of exponential, logarithmic, and inverse trigonometric functions. * Test: Differential Equations and Slope Fields Module 7: Applications of Integration Suggested Pace: 3 weeks * Application of integrals area and volume * Area of a region between two curves * Volume * Oral Review Discuss setup on a graphing calculator to find volumes for functions that cannot be integrated by hand. Students are required to be able to explain how the calculator is used to assist with the integration portion of solving a volume problem. Students have opportunity to demonstrate their solutions to other members of the class as well as the teacher using the whiteboard, application sharing of MathType and Graphmatica solutions, and the audio feature during this session. * Test Applications of Integration Module 8: Integration Techniques and L'Hopital's Rule Suggested Pace: 2 weeks * Techniques of Integration * Techniques for using Differentiation to find Limits * Basic rules of integration * Indeterminate forms and L Hopital s Rule Page 6 of 7

7 * Oral Review Students must verbally demonstrate the ability to use a calculator generated table to show limiting values of functions and comparative rates of growth of functions. * Test Integration Techniques Course Assessment and Participation Requirements: Besides engaging students in challenging curriculum, the course guides students to reflect on their learning and to evaluate their progress through a variety of assessments. Assessments can be in the form of self-checks, practice lessons, multiple choice questions, projects, oral assessments and discussions. Instructors evaluate progress and provide interventions through the variety of assessments built into a course, as well as through contact with the student in other venues. Page 7 of 7

AP Calculus BC Course Description

AP Calculus BC Course Description COURSE OUTLINE: The following topics define the AP Calculus BC course as it is taught over three trimesters, each consisting of twelve week grading periods. Limits and